I spent the last few days participating in the Canadian Conference in Computational Geometry, originally planned for Saskatoon but organized virtually instead.

The way the conference was organized was that (after the usual submission reviewing process) the accepted authors provided both a proceedings paper and a 10-15 minute talk video to the conference organizers. Participants were required to register, but with no registration fee, and were provided with links to the papers and talks (which are all still live on the conference program). Then, during the conference itself, live online Zoom sessions ran for only 2-3 hours daily, scheduled for 10AM-1PM Saskatoon time: very convenient for anywhere in North America or Europe, not so much for the participants in Iran, India, China, Japan, and Korea (all of which did have participants). The sessions included a daily 1-hour live invited talk, and question-and-answer sessions for the contributed works, in which we were shown a one-minute teaser for each video and then invited to ask questions of authors, at least one of whom was required to be present.

I think it all worked very well; so well, in fact, that during the business meeting there were calls for having at least some of the content similarly online so that people could participate remotely again. The ability to ask and answer questions either by live video on Zoom or through Zoom chat was useful, and used. Attendance was far above previous levels: 162 registrants, and over 120 unique participants at the most heavily attended of the two daily parallel sessions, compared to roughly 60 attendees each at the last two physical conferences. Despite the free registration, the conference organization was not without cost: they spent roughly \$1500 (Canadian) in video production costs and video conferencing fees, but this was more than made up for by institutional support for the conference, so they ended up running a surplus which may (if it can be kept) end up providing some float for future conference organizers.

There are two changes I would suggest for future events of this type:

• The contributed sessions were for a short enough time that holding them in parallel seemed unnecessary, and made it impossible to participate in all discussions. So this format may work better with less parallelism.

• The one-minute teaser videos were cut together by the conference organizers from the longer videos provided by the authors, but in some cases the pacing of the longer videos from which they were cut meant that these teaser videos could not clearly state the results of their papers. I think it would have been better to ask authors to provide these alongside the longer talk videos.

Some of the highlights of the event:

• Wednesday’s invited talk by Erik Demaine was a moving tribute to Godfried Toussaint and a survey of some of both Toussaint’s research, and Erik’s research on problems started by Toussaint, including convexifying polygons by flips, sona curves (the subject of one of my own contributions), the geometry of musical rhythms, and perhaps most importantly “supercollaboration”, the model of shared research and shared authorship developed by Toussaint at the Barbados research workshops and also used by Erik within many of his MIT classes. Erik’s talk was recorded and is now on YouTube; I hope the same will be true of the other two invited talks.

• In Wednesday’s contributed session on unfolding (in which I had two papers) I particularly liked Satyan Devadoss’s talk, “Nets of higher-dimensional cubes”. The main result is that if you unfold a hypercube, then no path of facets of the unfolded shape can contain a u-turn: if the path takes a step in one any coordinate direction, it cannot step in the opposite direction. This implies that all dual spanning trees unfold flat without self-intersection. The same property was known for all the Platonic solids, and Satyan can prove it for regular simplices in any dimension, but that still leaves several regular polytopes for which it is still open whether all unfoldings work: the cross-polytopes in all dimensions, and the three exceptional regular polytopes in four dimensions.

• Thursday’s invited talk was by Jeff Erickson, “Chasing puppies”. It was an entertaining presentation of an elegant topological proof of the following result: if you and a puppy can both move around a simple closed curve in the plane, with the puppy always moving along the curve to a local minimum of distance to you, then you can always find a path to follow that will bring you and the puppy to the same point.

• There wasn’t an official prize for best contributed presentation, but in the data structures session on Thursday, several comments nominated Don Sheehy as the unofficial winner, for a video artfully mixing live action with computer animations. His paper, “One-hop greedy permutations” concerned heuristics for improving the farthest-first traversal of a set of points by looking near each point as the sequence is constructed for a better point to use instead.

• There was an official best student paper award, and it went to Alejandro Flores Velazco for his paper “Social distancing is good for points too!” It concerns the problem of reducing the size of a data set while preserving the quality of nearest-neighbor classification using the reduced set. It proves that FCNN, which it calls the most popular heuristic for this problem, can produce significantly less-reduced outputs than some other proven heuristics, and shows how to modify FCNN to get a heuristic with guaranteed output quality.

• The third of the invited sessions was by Yusu Wang, newly moved from Ohio State to UC San Diego. She gave a nice introduction to combinatorial methods for reconstructing road networks (or other networks embedded into higher-dimensional geometry) from noisy samples of points on the network by combining discrete Morse theory to find the ridge lines of sample density with persistent homology to clean some of the noise from the data.

• The final technical component of the conference was an open problem session, also recorded and presumably to be uploaded at some point. Satyan posed his question on regular polytope unfolding there. Mike Paterson asked whether one can construct “Plato’s torus”, an embedded torus with six equilateral-triangle faces meeting at each vertex; in a blog post I made on this problem in 2009 I traced its history to Nick Halloway in 1997 but Mike says he discussed it already with Christopher Zeeman in the 1970s. Another problem that caught my attention asked for an algorithmic version of the polyhedral theorem of the three geodesics, the existence of a path across the surface of a convex polyhedron that stays straight across each edge or face of the polyhedron, and has at most $$\pi$$ surface angle on each side of it when it passes through a vertex. Again there’s some history here: Joe O’Rourke says he once mentioned the problem to Gromov, who said it was easy but unfortunately didn’t elaborate.

CCCG 2021 is planned for Halifax, colocated with WADS. One somewhat controversial issue is that the current plan is to have both conferences overlap for two days, with one overlap-free day for each conference at each end of the overlap period. But if both conferences are double-session, this means that participants can only choose one of four overlapping talks. At this point everyone is still hoping that events allow for a physical conference by then but that remains to be seen.